Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems

2025-12-24

AI Summary

This work addresses the quantum solution of nonlinear time-varying partial differential equations (PDEs), a long-standing challenge in quantum scientific computing. Methodologically, it introduces the first end-to-end physics-informed quantum algorithm, integrating quantum homotopy analysis with compact quantum finite differencing to construct high-dimensional linear embeddings. It defines, for the first time, a physics-adaptive nonlinearity metric—the homotopy Reynolds number $mathrm{Re}_H$—and establishes its quantitative relationships with integration step size, solution accuracy, and quantum resource requirements. Theoretical contributions include rigorous upper bounds on stability, error, quantum gate count, and query complexity; the encoding scheme is intrinsically adaptive to nonlinearity strength. Numerical experiments on the Burgers equation demonstrate asymptotic improvements—in matrix norm, condition number, simulation time, and accuracy—over existing quantum PDE solvers, significantly enhancing both practicality and scalability of quantum computation for real-world fluid dynamics problems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Quantum algorithms to integrate nonlinear PDEs governing flow problems are challenging to discover but critical to enhancing the practical usefulness of quantum computing. We present here a near-optimal, robust, and end-to-end quantum algorithm to solve time-dependent, dissipative, and nonlinear PDEs. We embed the PDEs in a truncated, high dimensional linear space on the basis of quantum homotopy analysis. The linearized system is discretized and integrated using finite-difference methods that use a compact quantum algorithm. The present approach can adapt its input to the nature of nonlinearity and underlying physics. The complexity estimates improve existing approaches in terms of scaling of matrix operator norms, condition number, simulation time, and accuracy. We provide a general embedding strategy, bounds on stability criteria, accuracy, gate counts and query complexity. A physically motivated measure of nonlinearity is connected to a parameter that is similar to the flow Reynolds number $Re_{ extrm{H}}$, whose inverse marks the allowed integration window, for given accuracy and complexity. We illustrate the embedding scheme with numerical simulations of a one-dimensional Burgers problem. This work shows the potential of the hybrid quantum algorithm for simulating practical and nonlinear phenomena on near-term and fault-tolerant quantum devices.

Problem

• Develops quantum algorithm for nonlinear PDEs
• Enhances quantum computing for flow problems
• Improves accuracy and complexity in simulations

Innovation

• Quantum homotopy embeds PDEs into linear space
• Compact quantum algorithm discretizes linearized system
• Adaptive input based on nonlinearity and physics